## Centroid, Orthocenter, Circumcenter & Incenter of a Triangle

**Centroid:**

- The centroid of a triangle is the point of intersection of medians. It divides medians in 2 : 1 ratio.
- IfA(x₁,y₁), B(x₂,y₂) and C(x₃,y₃) are vertices of triangle ABC, then coordinates of centroid is \(G=\left( \frac{{{x}_{1}}+{{x}_{2}}+{{x}_{3}}}{3},\,\frac{{{y}_{1}}+{{y}_{2}}+{{y}_{3}}}{3} \right)\).

**Incenter: **Point of intersection of angular bisectors

The **incenter** is the center of the incircle for a polygon or in sphere for a polyhedron (when they exist). The corresponding radius of the incircle or in sphere is known as the in radius. The **incenter** can be constructed as the intersection of angle bisectors coordinates of \(I=\left( \frac{a{{x}_{1}}+b{{x}_{2}}+c{{x}_{3}}}{a+b+c},\,\frac{a{{y}_{1}}+b{{y}_{2}}+c{{y}_{3}}}{a+b+c} \right)\)

Where a, b, c are sides of triangle ABC.

**Circumcenter:** The **circumcenter** is the center of a triangle’s circumcircle. It can be found as the intersection of the perpendicular bisectors

Point of intersection of perpendicular bisectors

Co-ordinates of circumcenter O is \(O=\left( \frac{{{x}_{1}}\sin 2A+{{x}_{2}}\sin 2B+{{x}_{3}}\sin 2C}{\sin 2A+\sin 2B+\sin 2C},\,\frac{{{y}_{1}}\sin 2A+{{y}_{2}}\sin 2B+{{y}_{3}}\sin 2C}{\sin 2A+\sin 2B+\sin 2C} \right)\)

**Orthocenter:** The **orthocenter** is the point where the three altitudes of a triangle intersect. A altitude is a perpendicular from a vertex to its opposite side

Point of intersection of altitudes of triangle ABC.

Coordinates of orthocenter H is \(H=\left( \frac{{{x}_{1}}\tan A+{{x}_{2}}\tan B+{{x}_{3}}\tan C}{\tan A+\tan B+\tan C},\,\frac{{{y}_{1}}\tan A+{{y}_{2}}\tan B+{{y}_{3}}\tan C}{\tan A+\tan B+\tan C} \right)\)

**Important points:**

- Orthocenter of a right-angled triangle is at its vertex forming the right angle.
- The orthocenter H, circumcenter O and centroid G of a triangle are collinear and G Divides H, O in ratio 2 : 1 i.e., HG: OG = 2 : 1